Finite Group Databasebeta

← #26224322

Subgroups — #26224322

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689475C211yesmaximal, characteristic
2C4×#13094024C211yesmaximal, characteristic
3#20244350C211yesmaximal, characteristic
4#20244189C211yesmaximal, characteristic
5#20244077C214yesmaximal
6#20243515C211yesmaximal, characteristic
7#20243221C214yesmaximal
8C4×#13093935C211yesmaximal, characteristic
9C4×C4⋊Q8×C47C211yesmaximal, characteristic
10C2×#13094024C2211yescharacteristic
11C2×C4×Dic94C2212yes—
12#15689475C418yescomplemented
13C2×#13093947C2214yes—
14C2×#13093947C2214yes—
15C2×#13093935C2212yes—
16C2×#13093935C2212yes—
17C2×C4×C47⋊C8C2211yescharacteristic
18C2×C4×C47⋊C8C2211yescharacteristic
19#15689029C2212yes—
20#15689021—21no—
21#15689021—21no—
22#15688993C2211yescharacteristic
23C4×C4⋊Dic47—21no—
24C4×C4⋊Dic47—21no—
25#15688885C2214yes—
26#15688885C2214yes—
27C2×C4⋊Q8×C47C2211yescharacteristic
28C2×C4×Q8×C47—21no—
29C47×#391—21no—
30C47×#389—21no—
31C47×#389C2212yes—
32C47×#384C2211yescharacteristic
33C4×C4⋊C4×C47—21no—
34C4×C4⋊C4×C47C2212yes—
35C43×C47C2211yescharacteristic
36C22×Dic94C2312yes—
37C22×C47⋊C8C2312yes—
38C22×C47⋊C8C2312yes—
39#13094024—22no—
40#13094024C2×C414yes—
41C4×Dic94—28no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47—21no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47C2312yes—
47C2×Dic47⋊C4—22no—
48C2×C4×Dic47—22no—
49#13093947—28no—
50#13093947C2×C4116yes—
51#13093935—24no—
52#13093935—24no—
53C4×C47⋊C8—22no—
54C4×C47⋊C8C2×C414yes—
55C47⋊#95—22no—
56C47⋊#95—22no—
57C22×Q8×C47—21no—
58C4⋊Q8×C47—22no—
59C4⋊Q8×C47C2×C414yes—
60C4×Q8×C47—44no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47C2312yes—
66C2×C4⋊C4×C47C2312yes—
67C2×C42×C47—22no—
68C2×C42×C47D411yescharacteristic
69C2×C42×C47D411yescharacteristic
70C2×C42×C47D411yescharacteristic
71C2×C42×C47D411yescharacteristic
72C2×C42×C47C2311yescharacteristic
73C2×C42×C47C2311yescharacteristic
74C2×C42×C47C2311yescharacteristic
75#95×C47—22no—
76#95×C47—22no—
77C2×Q8×C47—24no—
78C2×Q8×C47—42no—
79C2×Q8×C47—42no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2411yescharacteristic
89C22×Dic47—22no—
90C22×Dic47—22no—
91C2×Dic94—24no—
92C2×Dic94—24no—
93C2×Dic94—22no—
94C2×Dic94C22×C414yes—
95C4⋊C4×C47—24no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—24no—
98C4⋊C4×C47—44no—
99C4⋊C4×C47—42no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47C22×C414yes—
102C42×C47—22no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—22no—
106C42×C47—22no—
107C42×C47—44no—
108C42×C47C22×C412yes—
109C42×C47#4112yes—
110C42×C47#4114yes—
111C4⋊Dic47—24no—
112C4⋊Dic47—24no—
113C4⋊Dic47—24no—
114C4⋊Dic47—42no—
115C4⋊Dic47—42no—
116Dic47⋊C4—44no—
117C4×Dic47—44no—
118C2×C47⋊C8—22no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8C22×C414yes—
123C2×C47⋊C8#4114yes—
124C23×C47C22×D411yescharacteristic
125Q8×C47—48no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—24no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47C4×D412yes—
143C2×C4×C47C4×D412yes—
144C2×C4×C47#12412yes—
145C2×C4×C47#12412yes—
146C2×C4×C47AGL(1,ℤ/8)12yes—
147C2×C4×C47#13712yes—
148C2×C4×C47#13712yes—
149C2×C4×C47#13712yes—
150C2×C4×C47C23×C411yescharacteristic
151C2×C4×C47C2×#4112yes—
152C2×C4×C47C2×#4111yescharacteristic
153C2×C4×C47C2×#4112yes—
154C2×Dic47—24no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C2×Dic47—42no—
160Dic94—28no—
161Dic94—44no—
162C47⋊C8—24no—
163C47⋊C8—24no—
164#21647—471nomaximal, Sylow
165C22×C47C2×C4×D411yescharacteristic
166C22×C47C2×#12411yescharacteristic
167C22×C47#53211yescharacteristic
168C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
169C22×C47C2×#13711yescharacteristic
170C22×C47C2×#13711yescharacteristic
171C22×C47C2×#13711yescharacteristic
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—41no—
175C4×C47—41no—
176C4×C47—44no—
177C4×C47—82no—
178C4×C47#44014yes—
179C4×C47#44114yes—
180C4×C47C4×#4112yes—
181C4×C47#52912yes—
182Dic47—44no—
183Dic47—82no—
184C2×#487—471no—
185C4×C4⋊Q8—471no—
186C4×C4⋊Q8D4711yescharacteristic
187#1990—471no—
188#1956—471no—
189#1935—474no—
190#1834—471no—
191#1763—474no—
192C4×#97—471no—
193C2×C47C4×#12411yescharacteristic
194C2×C47C2×#44011yescharacteristic
195C2×C47C2×#44111yescharacteristic
196C2×C47#313911yescharacteristic
197C2×C47#314111yescharacteristic
198C2×C47#318211yescharacteristic
199C2×C47#318511yescharacteristic
200C2×C4⋊Q8—471no—
201C2×C4⋊Q8C2×D4711yescharacteristic
202C2×C4×Q8—21no—
203C2×C4×Q8—472no—
204#487—478no—
205C2×#103—474no—
206C2×#103—474no—
207C2×#97—472no—
208C2×#97—472no—
209C2×C4×C8—471no—
210C2×C4×C8—471no—
211#391—21no—
212#391—472no—
213#389—21no—
214#389—941no—
215#389—941no—
216#389C2×D4712yes—
217#384—471no—
218#384C2×D4711yescharacteristic
219C4×C4⋊C4—21no—
220C4×C4⋊C4—941no—
221C4×C4⋊C4—941no—
222C4×C4⋊C4C2×D4712yes—
223C43C2×D4711yescharacteristic
224#340—474no—
225#340—474no—
226C47#2164711yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—472no—
229C22×C8—472no—
230C22×C8—472no—
231C4⋊Q8—22no—
232C4⋊Q8—474no—
233C4⋊Q8—942no—
234C4⋊Q8C4×D4714yes—
235C4×Q8—44no—
236C4×Q8—948no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—472no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—941no—
244C2×C4⋊C4—942no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4C22×D4712yes—
248C2×C4⋊C4C22×D4712yes—
249C2×C42—22no—
250C2×C42—942no—
251C2×C42C22⋊D4711yescharacteristic
252C2×C42C22⋊D4711yescharacteristic
253C2×C42C22⋊D4711yescharacteristic
254C2×C42C22⋊D4711yescharacteristic
255C2×C42C22×D4711yescharacteristic
256C2×C42C22×D4711yescharacteristic
257C2×C42C22×D4711yescharacteristic
258#103—4716no—
259#103—948no—
260#97—944no—
261#97—944no—
262C4×C8—474no—
263C4×C8—942no—
264#95—22no—
265#95—22no—
266#95—942no—
267#95—942no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—474no—
272C2×Q8—942no—
273C2×Q8—944no—
274C2×Q8—944no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—942no—
278C22×C4—942no—
279C22×C4C2×C22⋊D4711yescharacteristic
280C22×C4C2×C22⋊D4711yescharacteristic
281C22×C4C2×C22⋊D4711yescharacteristic
282C22×C4C2×C22⋊D4711yescharacteristic
283C22×C4C2×C22⋊D4711yescharacteristic
284C22×C4C2×C22⋊D4711yescharacteristic
285C22×C4C23×D4711yescharacteristic
286C2×C8—474no—
287C2×C8—474no—
288C2×C8—942no—
289C2×C8—942no—
290C2×C8—942no—
291C2×C8—942no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—944no—
299C4⋊C4—944no—
300C4⋊C4—944no—
301C4⋊C4—1884no—
302C4⋊C4—1882no—
303C4⋊C4—1882no—
304C4⋊C4C2×C4×D4714yes—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—44no—
311C42—1884no—
312C42C2×C4×D4712yes—
313C42#1062535012yes—
314C42#1062535414yes—
315C23C22×C22⋊D4711yescharacteristic
316Q8—48no—
317Q8—948no—
318Q8—1884no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—24no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—944no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4C4×C22⋊D4712yes—
342C2×C4C4×C22⋊D4712yes—
343C2×C4#1309401312yes—
344C2×C4#1309401312yes—
345C2×C4#1309405012yes—
346C2×C4#1309405412yes—
347C2×C4#1309405412yes—
348C2×C4#1309405512yes—
349C2×C4C22×C4×D4711yescharacteristic
350C2×C4#41×D4712yes—
351C2×C4C2×#1062535011yescharacteristic
352C2×C4C2×#1062535412yes—
353C8—944no—
354C8—944no—
355C22C2×C4×C22⋊D4711yescharacteristic
356C22C2×#1309401311yescharacteristic
357C22#1568966911yescharacteristic
358C22C2×#1309405011yescharacteristic
359C22C2×#1309405411yescharacteristic
360C22C2×#1309405411yescharacteristic
361C22C2×#1309405511yescharacteristic
362C4—22no—
363C4—22no—
364C4—44no—
365C4—41no—
366C4—41no—
367C4—82no—
368C4—3762no—
369C4—1884no—
370C4#1568922314yes—
371C4#1568922814yes—
372C4C4×#1062535412yes—
373C4#1568966012yes—
374C2C4×#1309401311yescharacteristic
375C2C2×#1568922311yescharacteristic
376C2C2×#1568922811yescharacteristic
377C2#2025377011yescharacteristic
378C2#2025378411yescharacteristic
379C2#2025410211yescharacteristic
380C2#2025412911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×C4⋊Q8×C47C2
socle, prime radicalC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenterC4×C4⋊Q8D47
hypocenterC47#21647
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #15689475C4semidirect
226. C47#21647semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156894752
C4×#130940242
#202443502
#202441892
#202440772
#202435152
#202432212
C4×#130939352
C4×C4⋊Q8×C472
#2164747